A Cycloid is a Tautochrone curve, meaning all points take the same time to reach the bottom regardless of initial heightpic.twitter.com/sX94kTPWll
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Is there some transparent reason why the two properties hold? A way to see that a brachistochrone must be tautochronous without computing it?
I think if there were it wouldn’t have been an open problem for so long and took Newton inventing calculus of variations
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AFAIK Huygens first solved the tautochrone problem in 1659, whereas the brachistochrone problem wasn’t even proposed by Bernoulli until 1696, after which Newton solved it in a night. So I don’t think history rules out a simple way to see that the two shapes have to be the same.
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However, the two curves are different in almost every situation *except* that of a constant acceleration (linear potential). So, in this sense, it does seem like a miracle that these two happen to be the same curve.
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